Strange metals and

black holes Subir Sachdev

Dynamics and Disorder in Quantum Many Body Systems Far from Equilibrium

Les Houches Summer School, August 19-21, 2019

HARVARD

Remarkable recent observation of‘Planckian’ strange metal transport in cuprates,pnictides, magic-angle graphene, andultracold atoms: the resistivity, ⇢, is

⇢ =m⇤

ne21

⌧

with a universal scattering rate

1

⌧⇡ kBT

~ ,

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REPORT◥

QUANTUM SIMULATION

Bad metallic transport in a cold atomFermi-Hubbard systemPeter T. Brown1, Debayan Mitra1, Elmer Guardado-Sanchez1, Reza Nourafkan2,Alexis Reymbaut2, Charles-David Hébert2, Simon Bergeron2, A.-M. S. Tremblay2,3,Jure Kokalj4,5, David A. Huse1, Peter Schauß1*, Waseem S. Bakr1†

Strong interactions in many-body quantum systems complicate the interpretation ofcharge transport in such materials. To shed light on this problem, we study transport in aclean quantum system: ultracold lithium-6 in a two-dimensional optical lattice, a testingground for strong interaction physics in the Fermi-Hubbard model. We determine thediffusion constant by measuring the relaxation of an imposed density modulation andmodeling its decay hydrodynamically. The diffusion constant is converted to a resistivity byusing the Nernst-Einstein relation. That resistivity exhibits a linear temperaturedependence and shows no evidence of saturation, two characteristic signatures of a badmetal. The techniques we developed in this study may be applied to measurements ofother transport quantities, including the optical conductivity and thermopower.

In conventional materials, charge is carriedby quasiparticles and conductivity is under-stood as a current of these charge carriersdeveloped in response to an external field.For the conductivity to be finite, the charge

carriers must be able to relax their momentumthrough scattering. The Boltzmann kinetic equa-tion in conjunction with Fermi liquid theoryprovides a detailed description of transport inconventionalmaterials, including two trademarksof resistivity. The first is the Fermi liquid predictionthat the temperature (T)–dependent resistivity r(T)should scale like T2 at low temperature (1). Thesecond is that the resistivity should not exceedamaximum value rmax, obtained from the Druderelation assuming the Mott-Ioffe-Regel (MIR)limit, which states that the mean free path of aquasiparticle cannot be less than the lattice spacing(2, 3). This resistivity bound itself is sometimesreferred to as the MIR limit.Strong interactions can, however, lead to a

breakdown of Fermi liquid theory. One signal ofthis breakdown is anomalous scaling of r withtemperature, including the linear scaling ob-served in the strange metal state of the cuprates(4) and other anomalous scalings in d- and f-electron materials (5). Another is the violation ofthe resistivity bound r < rmax, which is observed

in a wide variety of materials (6). Additionally,interactions may lead to a situation where themomentum relaxation rate alone does not de-termine the conductivity, in contrast to thesemiclassical Drude formula, generalizationsof which hold for a large class of systems calledcoherent metals (7). Approaches introduced tounderstand these anomalous behaviors includehidden Fermi liquids (8), marginal Fermi liquids(9), proximity to quantum critical points (10) andassociated holographic approaches (11), andmany numerical studies of model systems,most notably the Hubbard model (12) and thet − J model (13).Disentangling strong interaction physics from

other effects, such as impurities and electron-phonon coupling, is difficult in real materials.Cold atom systems are free of these complica-tions, but transport experiments are challengingbecause of the finite and isolated nature of thesesystems. Most fermionic charge transport ex-periments have focused on studying either massflow through optically structured mesoscopicdevices (14–17) or bulk transport in lattice sys-tems (18–22). In this study, we explored bulktransport in a Fermi-Hubbard system by study-ing charge diffusion, which is a microscopicprocess related to conductivity through theNernst-Einstein equation s = ccD, where D isthe diffusion constant and cc ¼ @n

@m

! "jT is the

compressibility. This requires only the assumptionof linear response and the absence of thermo-electric coupling (23) anddoes not rest on assump-tions concerning quasiparticles.We realized the two-dimensional Fermi-Hubbard

model by using a degenerate spin-balancedmix-ture of two hyperfine ground states of 6Li in anoptical lattice (24). Our lattice beams producea harmonic trapping potential, which leads to avarying atomic density in the trap. To obtain a

system with uniform density, we flatten ourtrapping potential over an elliptical region ofmean diameter 30 sites by using a repulsivepotential created with a spatial light modulator.We superimpose an additional sinusoidal po-tential that varies slowly along one direction ofthe lattice with a controllable wavelength (Fig. 1,A and B). By adiabatically loading the gas intothese potentials, we prepare a Hubbard systemin thermal equilibrium with a small-amplitude(typically 10%) sinusoidal density modulation.The average density in the region with the flat-tened potential is the same with and without thesinusoidal potential. Next, we suddenly turn offthe added sinusoidal potential and observe thedecay of the density pattern versus time (Fig. 1,C and D), always keeping the optical lattice atfixed intensity. We measure the density of asingle spin component, hn↑i, by using techniquesdescribed in (24), giving us access to the totaldensity through hni ¼ h2n↑i.Wework at average total densityhni ¼ 0:82ð2Þ.

This value is close to a conjectured quantumcritical point in the Hubbard model (25). Ourlattice depth is 6.9(2) ER, where ER is the latticerecoil energy and ER/h = 14.66 kHz, leading toa tunneling rate of t/h = 925(10) Hz. Here h isPlanck’s constant. We adjust the scatteringlength, as = 1070(10)ao, by working at a mag-netic bias field of 616.0(2) G, in the vicinity ofthe Feshbach resonance centered near 690 G.These parameters lead to an on-site interaction–to–tunneling ratio U/t = 7.4(8), which is in thestrong-interaction regime and close to the valuethatmaximizes antiferromagnetic correlations athalf-filling (26).We observe the decay of the initial sinusoidal

density pattern over a period of a few tunnelingtimes. The short time scale ensures that theobserved dynamics are not affected by the in-homogeneous density outside of the centralflattened region of the trap. To obtain betterstatistics, we apply the sinusoidal modulationalong one dimension and average along theother direction (Fig. 1, A and C). We fit theaverage modulation profile to a sinusoid, wherethe phase and frequency are fixed by the initialpattern (Fig. 2A). The time dependence of theamplitude of the sinusoid quantifies the decayof the density modulation (Fig. 2B). Our experi-mental technique is analogous to that ofHild et al.(27), who studied the decay of a sinusoidallymodulated spin pattern in a bosonic system.The decay of the sinusoidal density pattern

versus the wavelength l of the modulation be-comes consistent with diffusive transport at longwavelengths. In diffusive transport, the ampli-tude of a density pattern at wave vector k = 2p/lwill decay exponentially with time constant t =1/Dk2, where D is the diffusion constant. We ob-serve exponentially decaying amplitudes withdiffusive scaling for wavelengths longer than15 sites. However, the decay curves are flat atearly times, showing clear deviation from ex-ponential decay. For short wavelengths, we ob-serve deviations from diffusive behavior in theform of underdamped oscillations, which can be

RESEARCH

Brown et al., Science 363, 379–382 (2019) 25 January 2019 1 of 4

1Department of Physics, Princeton University, Princeton, NJ08544, USA. 2Département de Physique, Institut Quantique,and Regroupement Québécois sur les Matériaux de Pointe,Université de Sherbrooke, Sherbrooke, Québec J1K 2R1,Canada. 3Canadian Institute for Advanced Research, Toronto,Ontario M5G 1Z8, Canada. 4Faculty of Civil and GeodeticEngineering, University of Ljubljana, SI-1000 Ljubljana,Slovenia. 5Jožef Stefan Institute, Jamova 39, SI-1000Ljubljana, Slovenia.*Present address: Department of Physics, University of Virginia,Charlottesville, VA 22904, USA.†Corresponding author. Email: wbakr@princeton.edu

on February 13, 2019

http://science.sciencemag.org/

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Strange metal in magic-angle graphene with near Planckian dissipation

Yuan Cao,1, ⇤ Debanjan Chowdhury,1, ⇤ Daniel Rodan-Legrain,1 Oriol Rubies-Bigorda,1

Kenji Watanabe,2 Takashi Taniguchi,2 T. Senthil,1, † and Pablo Jarillo-Herrero1, †

1Department of Physics, Massachusetts Institute of Technology, Cambridge Massachusetts 02139, USA.

2National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan.

(Dated: January 15, 2019)

Recent experiments on magic angle twisted bilayer graphene have discovered correlated insulating

behavior and superconductivity at a fractional filling of an isolated narrow band. In this paper we show

that magic angle bilayer graphene exhibits another hallmark of strongly correlated systems — a broad

regime of T�linear resistivity above a small, density dependent, crossover temperature— for a range of

fillings near the correlated insulator. We also extract a transport “scattering rate”, which satisfies a near

Planckian form that is universally related to the ratio of (kBT/~). Our results establish magic angle bilayer

graphene as a highly tunable platform to investigate strange metal behavior, which could shed light on

this mysterious ubiquitous phase of correlated matter.

A panoply of strongly correlated materials have metallicparent states that display properties at odds with the expecta-tions in a conventional Fermi liquid and are marked by the ab-sence of coherent quasiparticle excitations. Some well knownfamilies of materials, such as the ruthenates [1, 2], cobal-tates [3, 4] and a subset of the iron-based superconductors [5],show non-Fermi liquid (NFL) behavior over a broad inter-mediate range of temperatures, with a crossover to a conven-tional Fermi liquid below an emergent low-energy scale, Tcoh,at which coherent electronic quasiparticles emerge as well-defined excitations. Even more striking examples of NFLbehavior are observed in the hole-doped cuprates [6, 7] andcertain quantum critical heavy-fermion compounds [8] wherethe incoherent features appear to survive down to the lowestmeasurable temperatures, i.e. Tcoh ! 0, when superconduc-tivity is suppressed externally. In the incoherent regime, all ofthese materials in spite of being microscopically distinct havea resistivity, ⇢(T ) ⇠ T , and exhibit a number of anomalousfeatures [8–11] clearly indicating the absence of sharp elec-tronic quasiparticles. In strongly interacting non-quasiparticlesystems, it has been conjectured [12] that transport “scatteringrates” (�) satisfy a universal ‘Planckian’ bound � . O(kBT/~)at a temperature T . It is however worth noting that in a NFLthere is in general no clear definition for �; the scattering ratesdefined through di↵erent measurements, such as dc and opti-cal conductivity, need not be identical [13]. One of the mostsurprising aspects of incoherent transport in these systems, inspite of these subtleties, is that � extracted from the dc resis-tivity (through a procedure specified in Ref. [14]) appears tosatisfy a universal form � = CkBT/~with C a number of order1 [14, 15].

Recent experiments [16, 17] have reported the discoveryof a correlation driven insulator at fractional fillings (with re-spect to a fully filled isolated band) in magic-angle bilayergraphene (MABLG). In MABLG, the relative rotation be-tween two sheets of graphene generates a moire pattern (Fig.1a) with a periodicity that is much larger than the underly-ing interatomic distances in graphene. The theoretically esti-mated electronic bandwidth, W, is strongly renormalized near

these small magic-angles [18–20]; then the strength of the typ-ical Coulomb interactions, U, becomes at least comparable to(if not greater than) the bandwidth, U & W. Investigatingthe properties of MABLG as a function of temperature andcarrier density with unprecedented tunability in a controlledsetup can lead to new insights into the nature of electronictransport in other low-dimensional strongly correlated metal-lic systems.

For our experiments, we have fabricated multiple high-quality encapsulated MABLG devices (see Fig. 1a) usingthe ‘tear and stack’ technique [21, 22]. As reported earlier[16, 22], we obtain band-insulators with a large gap nearn ⇡ ±ns, where n is the carrier density tuned externally byapplying a gate voltage and ns corresponds to four electronsper moire unit cell. On the other hand, correlation driven in-sulators with much smaller gap-scales [16, 23] appear nearn ⇡ ±ns/2; we denote these fillings as ⌫ = ±2 from nowon (the fully filled band corresponds to ⌫ = +4). Dop-ing away from these correlated insulators by an additionalamount, �, with holes (⌫ = ±2 � �) and electrons (⌫ = ±2 + �)leads to superconductivity (SC) [17, 23]. The superconduct-ing transition temperature (Tc) measured relative to the Fermi-energy ("F), as inferred from low temperature quantum oscil-lations measurements [17], is high, with the largest value of(Tc/"F) ⇠ 0.07 � 0.08, indicating strong coupling supercon-ductivity [17]. A schematic ⌫�T phase-diagram for MABLGis shown in Fig.1b.

In this work, we investigate the transport phenomenologyof the metallic states in MABLG as a function of increasingtemperature over a large range of externally tuned fillings. Wehave analyzed the temperature dependence of the longitudinalDC resistivity, ⇢(T ), for a number of devices (MA1 - MA6)over a wide range of fillings, the results of which appear tobe qualitatively similar across devices. In order to highlightthe universal aspects of the behavior, both within and acrossdi↵erent samples, we show the temperature dependent tracesof ⇢(T ) for a range of di↵erent ⌫ in two devices MA1 andMA4 in Fig.1c and Fig.1d respectively. The range of ⌫ chosenfor this purpose is shown as a color-bar in Fig.1b; see caption

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Planckian dissipation and scale invariance in a quantum-critical disordered pnictide

Yasuyuki Nakajima,1, 2 Tristin Metz,2 Christopher Eckberg,2 Kevin Kirshenbaum,2 Alex Hughes,2

Renxiong Wang,2 Limin Wang,2 Shanta R. Saha,2 I-Lin Liu,2, 3, 4 Nicholas P. Butch,2, 4

Zhonghao Liu,5, 6 Sergey V. Borisenko,5 Peter Y. Zavalij,7 and Johnpierre Paglione2, 8

1Department of Physics, University of Central Florida, Orlando, Florida 328162Center for Nanophysics and Advanced Materials, Department of Physics,

University of Maryland, College Park, Maryland 207423Chemical Physics Department, University of Maryland, College Park, Maryland 20742

4NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 208995IFW-Dresden, Helmholtzstraße 20, 01171 Dresden, Germany

6Shanghai Institute of Microsystem and Information Technology,Chinese Academy of Sciences, Shanghai 200050, China

7Department of Chemistry, University of Maryland, College Park, Maryland 207428The Canadian Institute for Advanced Research, Toronto, ON M5G 1Z8, Canada

(Dated: February 5, 2019)

Quantum-mechanical fluctuations between competing phases at T = 0 induce exotic finite-temperature collective excitations that are not described by the standard Landau Fermi liquidframework [1–4]. These excitations exhibit anomalous temperature dependences, or non-Fermi liq-uid behavior, in the transport and thermodynamic properties [5] in the vicinity of a quantumcritical point, and are often intimately linked to the appearance of unconventional Cooper pairingas observed in strongly correlated systems including the high-Tc cuprate and iron pnictide supercon-ductors [6, 7]. The presence of superconductivity, however, precludes direct access to the quantumcritical point, and makes it difficult to assess the role of quantum-critical fluctuations in shapinganomalous finite-temperature physical properties, such as Planckian dissipation !/τp = kBT [8–10].Here we report temperature-field scale invariance of non-Fermi liquid thermodynamic, transportand Hall quantities in a non-superconducting iron-pnictide, Ba(Fe1/3Co1/3Ni1/3)2As2, indicative ofquantum criticality at zero temperature and zero applied magnetic field. Beyond a linear in temper-ature resistivity, the hallmark signature of strong quasiparticle scattering, we find a more universalPlanck-limited scattering rate that obeys a scaling relation between temperature and applied mag-netic fields down to the lowest energy scales. Together with the emergence of hole-like carriers closeto the zero-temperature and zero-field limit, the scale invariance, isotropic field response and lackof applied pressure sensitivity point to the realization of a novel quantum fluid predicted by theholographic correspondence [11] and born out of a unique quantum critical system that does notdrive a pairing instability.

Non-Fermi liquid (NFL) behavior ubiquitously ap-pears in iron-based high-temperature superconductorswith a novel type of superconducting pairing symme-try driven by interband repulsion [7, 12]. The putativepairing mechanism is thought to be associated with thetemperature-doping phase diagram, bearing striking re-semblance to cuprate and heavy-fermion superconductors[13, 14]. In iron-based superconductors, the supercon-ducting phase appears to be centered around the point ofsuppression of antiferromagnetic (AFM) and orthorhom-bic structural order [12]. Close to the boundary betweenAFM order and superconductivity, the exotic metallicregime emerges in the normal state. In addition to theAFM order, the presence of an electronic nematic phaseabove the structural transition complicates the under-standing of the SC and NFL behavior [15–18]. More-over, the robust superconducting phase prohibits inves-tigations of zero-temperature limit normal state physicalproperties associated with the quantum critical (QC) in-stability due to the extremely high upper critical fields.

While AFM spin fluctuations are widely believed toprovide the pairing glue in the iron-pnictides, other mag-

netic interactions are prevalent in closely related ma-terials, such as the cobalt-based oxypnictides LaCoOX(X=P, As) [19], which exhibit ferromagnetic (FM) or-ders, and Co-based intermetallic arsenides with coexist-ing FM and AFM spin correlations [20–22]. For instance,a strongly enhanced Wilson ratio RW of ∼ 7-10 at 2 K[23] and violation of the Koringa law [20–22] suggestproximity to a FM instability in BaCo2As2. BaNi2As2,on the other hand, seems to be devoid of magnetic or-der [24] and rather hosts other ordering instabilities inboth structure and charge [25]. Confirmed by extensivestudy, Fe, Co, and Ni have the same 2+ oxidation statein the tetragonal ThCr2Si2 structure, thus adding one delectron- (hole-) contribution by Ni (Fe) substitution forCo in BaCo2As2 [26–29], and thereby modifying the elec-tronic structure subtly but significantly enough to tune inand out of different ground states and correlation types.Utilizing this balance, counter-doping a system to achievethe same nominal d electron count as BaCo2As2 can re-alize a unique route to the same nearly FM system whiledisrupting any specific spin correlation in the system.

Here, we utilize this approach to stabilize a novel

arXiv:1902.01034v1 [cond-mat.str-el] 4 Feb 2019

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LETTERShttps://doi.org/10.1038/s41567-018-0334-2

1Institut Quantique, Département de Physique & RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada. 2SPEC, CEA, CNRS-UMR 3680, Université Paris-Saclay, Gif sur Yvette Cedex, France. 3Laboratoire National des Champs Magnétiques Intenses (CNRS, EMFL, INSA, UJF, UPS), Toulouse, France. 4AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, Poland. 5Laboratoire de Physique des Solides, Université Paris-Sud, Université Paris-Saclay, CNRS UMR 8502, Orsay, France. 6Canadian Institute for Advanced Research, Toronto, Ontario, Canada. *e-mail: louis.taillefer@usherbrooke.ca; cyril.proust@lncmi.cnrs.fr

The perfectly linear temperature dependence of the electrical resistivity observed as T!→ !0 in a variety of metals close to a quantum critical point1–4 is a major puzzle of condensed-mat-ter physics5. Here we show that T-linear resistivity as T!→ !0 is a generic property of cuprates, associated with a universal scattering rate. We measured the low-temperature resistivity of the bilayer cuprate Bi2Sr2CaCu2O8+δ and found that it exhib-its a T-linear dependence with the same slope as in the single-layer cuprates Bi2Sr2CuO6+δ (ref.!6), La1.6−xNd0.4SrxCuO4 (ref.!7) and La2−xSrxCuO4 (ref.!8), despite their very different Fermi surfaces and structural, superconducting and magnetic prop-erties. We then show that the T-linear coefficient (per CuO2 plane), A1

□, is given by the universal relation A1□TF!= !h/2e2,

where e is the electron charge, h is the Planck constant and TF is the Fermi temperature. This relation, obtained by assum-ing that the scattering rate 1/τ of charge carriers reaches the Planckian limit9,10, whereby ħ/τ!= !kBT, works not only for hole-doped cuprates6–8,11,12 but also for electron-doped cuprates13,14, despite the different nature of their quantum critical point and strength of their electron correlations.

In conventional metals, the electrical resistivity ρ(T) normally varies as T2 in the limit T → 0, where electron–electron scattering dominates, in accordance with Fermi-liquid theory. However, close to a quantum critical point (QCP) where a phase of antiferromag-netic order ends, ρ(T) ~ Tn, with n < 2.0. Most striking is the obser-vation of a perfectly linear T dependence ρ(T) = ρ0 + A1T as T → 0 in several very different materials, when tuned to their magnetic QCP; for example, the quasi-one-dimensional (1D) organic conductor (TMTSF)2PF6 (ref. 4), the quasi-2D ruthenate Sr3Ru2O7 (ref. 3) and the 3D heavy-fermion metal CeCu6 (ref. 1). This T-linear resistiv-ity as T → 0 has emerged as one of the major puzzles in the physics of metals5, and while several theoretical scenarios have been pro-posed15, no compelling explanation has been found.

In cuprates, a perfect T-linear resistivity as T → 0 has been observed (once superconductivity is suppressed by a magnetic field) in two closely related electron-doped materials, Pr2−xCexCuO4±δ (PCCO)2,16,17 and La2−xCexCuO4 (LCCO)13,14, and in three hole-doped materials: Bi2Sr2CuO6+δ (ref. 6), La2−xSrxCuO4 (LSCO)8 and La1.6−xNd0.4SrxCuO4 (Nd-LSCO)7,11,12. On the electron-doped side, T-linear resistivity is seen just above the QCP16 where antiferromag-netic order ends18 as a function of x, and as such it may not come as a surprise. On the hole-doped side, however, the doping values

where ρ(T) = ρ0 + A1T as T → 0 are very far from the QCP where long-range antiferromagnetic order ends (pN ~ 0.02); for example, at p = 0.24 in Nd-LSCO (Fig. 1a) and in the range p = 0.21–0.26 in LSCO (Fig. 1b). Instead, these values are close to the critical dop-ing where the pseudogap phase ends (that is, at p* = 0.23 ± 0.01 in Nd-LSCO (ref. 11) and at p* ~ 0.18–0.19 in LSCO (ref. 8)), where the role of antiferromagnetic spin fluctuations is not clear. In Bi2201, p* is farther still (see Supplementary Section 10).

To make progress, several questions must be answered. Is T-linear resistivity as T → 0 in hole-doped cuprates limited to single-layer materials with low Tc, or is it generic? Why is ρ(T) = ρ0 + A1T as T → 0 seen in LSCO over an anomalously wide doping range8? Is there a common mechanism linking cuprates to the other metals where ρ ~ T as T → 0?

To establish the universal character of T-linear resistivity in cuprates, we have turned to Bi2Sr2CaCu2O8+δ (Bi2212). While Nd-LSCO and LSCO have essentially the same single electron-like diamond-shaped Fermi surface at p > p* (refs 19,20), Bi2212 has a very different Fermi surface, consisting of two sheets, one of which is also diamond-like at p > 0.22, but the other is much more circular21 (see Supplementary Section 1). Moreover, the structural, magnetic and superconducting properties of Bi2212 are very different to those of Nd-LSCO and LSCO: a stronger 2D character, a larger gap to spin excitations, no spin-density-wave order above p ~ 0.1 and a much higher superconducting Tc.

We measured the resistivity of Bi2212 at p = 0.23 by sup-pressing superconductivity with a magnetic field of up to 58 T. At p = 0.23, the system is just above its pseudogap critical point (p* = 0.22 (ref. 22); see Supplementary Section 2). Our data are shown in Fig. 2. The raw data at H = 55 T reveal a perfectly linear T dependence of ρ(T) down to the lowest accessible temperature (Fig. 1a). Correcting for the magnetoresistance (see Methods and Supplementary Section 3), as was done for LSCO (ref. 8), we find that the T-linear dependence of ρ(T) seen in Bi2212 at H = 0 from T ~ 120 K down to Tc simply continues to low temperature, with the same slope A1 = 0.62 ± 0.06 μ Ω cm K−1 (Fig. 2b). Measured per CuO2 plane, this gives A1

□ ≡ A1/d = 8.0 ± 0.9 Ω K−1, where d is the (aver-age) separation between CuO2 planes. Remarkably, this is the same value, within error bars, as measured in Nd-LSCO at p = 0.24, where A1

□ = 7.4 ± 0.8 Ω K−1 (see Table 1).The observation of T-linear resistivity in those two cuprates

shows that it is robust against changes in the shape, topology and

Universal T-linear resistivity and Planckian dissipation in overdoped cupratesA. Legros1,2, S. Benhabib3, W. Tabis3,4, F. Laliberté" "1, M. Dion1, M. Lizaire1, B. Vignolle3, D. Vignolles" "3, H. Raffy5, Z. Z. Li5, P. Auban-Senzier5, N. Doiron-Leyraud1, P. Fournier1,6, D. Colson2, L. Taillefer" "1,6* and C. Proust" "3,6*

NATURE PHYSICS | VOL 15 | FEBRUARY 2019 | 142–147 | www.nature.com/naturephysics142

LETTERShttps://doi.org/10.1038/s41567-018-0334-2

1Institut Quantique, Département de Physique & RQMP, Université de Sherbrooke, Sherbrooke, Québec, Canada. 2SPEC, CEA, CNRS-UMR 3680, Université Paris-Saclay, Gif sur Yvette Cedex, France. 3Laboratoire National des Champs Magnétiques Intenses (CNRS, EMFL, INSA, UJF, UPS), Toulouse, France. 4AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, Poland. 5Laboratoire de Physique des Solides, Université Paris-Sud, Université Paris-Saclay, CNRS UMR 8502, Orsay, France. 6Canadian Institute for Advanced Research, Toronto, Ontario, Canada. *e-mail: louis.taillefer@usherbrooke.ca; cyril.proust@lncmi.cnrs.fr

The perfectly linear temperature dependence of the electrical resistivity observed as T!→ !0 in a variety of metals close to a quantum critical point1–4 is a major puzzle of condensed-mat-ter physics5. Here we show that T-linear resistivity as T!→ !0 is a generic property of cuprates, associated with a universal scattering rate. We measured the low-temperature resistivity of the bilayer cuprate Bi2Sr2CaCu2O8+δ and found that it exhib-its a T-linear dependence with the same slope as in the single-layer cuprates Bi2Sr2CuO6+δ (ref.!6), La1.6−xNd0.4SrxCuO4 (ref.!7) and La2−xSrxCuO4 (ref.!8), despite their very different Fermi surfaces and structural, superconducting and magnetic prop-erties. We then show that the T-linear coefficient (per CuO2 plane), A1

□, is given by the universal relation A1□TF!= !h/2e2,

where e is the electron charge, h is the Planck constant and TF is the Fermi temperature. This relation, obtained by assum-ing that the scattering rate 1/τ of charge carriers reaches the Planckian limit9,10, whereby ħ/τ!= !kBT, works not only for hole-doped cuprates6–8,11,12 but also for electron-doped cuprates13,14, despite the different nature of their quantum critical point and strength of their electron correlations.

In conventional metals, the electrical resistivity ρ(T) normally varies as T2 in the limit T → 0, where electron–electron scattering dominates, in accordance with Fermi-liquid theory. However, close to a quantum critical point (QCP) where a phase of antiferromag-netic order ends, ρ(T) ~ Tn, with n < 2.0. Most striking is the obser-vation of a perfectly linear T dependence ρ(T) = ρ0 + A1T as T → 0 in several very different materials, when tuned to their magnetic QCP; for example, the quasi-one-dimensional (1D) organic conductor (TMTSF)2PF6 (ref. 4), the quasi-2D ruthenate Sr3Ru2O7 (ref. 3) and the 3D heavy-fermion metal CeCu6 (ref. 1). This T-linear resistiv-ity as T → 0 has emerged as one of the major puzzles in the physics of metals5, and while several theoretical scenarios have been pro-posed15, no compelling explanation has been found.

In cuprates, a perfect T-linear resistivity as T → 0 has been observed (once superconductivity is suppressed by a magnetic field) in two closely related electron-doped materials, Pr2−xCexCuO4±δ (PCCO)2,16,17 and La2−xCexCuO4 (LCCO)13,14, and in three hole-doped materials: Bi2Sr2CuO6+δ (ref. 6), La2−xSrxCuO4 (LSCO)8 and La1.6−xNd0.4SrxCuO4 (Nd-LSCO)7,11,12. On the electron-doped side, T-linear resistivity is seen just above the QCP16 where antiferromag-netic order ends18 as a function of x, and as such it may not come as a surprise. On the hole-doped side, however, the doping values

where ρ(T) = ρ0 + A1T as T → 0 are very far from the QCP where long-range antiferromagnetic order ends (pN ~ 0.02); for example, at p = 0.24 in Nd-LSCO (Fig. 1a) and in the range p = 0.21–0.26 in LSCO (Fig. 1b). Instead, these values are close to the critical dop-ing where the pseudogap phase ends (that is, at p* = 0.23 ± 0.01 in Nd-LSCO (ref. 11) and at p* ~ 0.18–0.19 in LSCO (ref. 8)), where the role of antiferromagnetic spin fluctuations is not clear. In Bi2201, p* is farther still (see Supplementary Section 10).

To make progress, several questions must be answered. Is T-linear resistivity as T → 0 in hole-doped cuprates limited to single-layer materials with low Tc, or is it generic? Why is ρ(T) = ρ0 + A1T as T → 0 seen in LSCO over an anomalously wide doping range8? Is there a common mechanism linking cuprates to the other metals where ρ ~ T as T → 0?

To establish the universal character of T-linear resistivity in cuprates, we have turned to Bi2Sr2CaCu2O8+δ (Bi2212). While Nd-LSCO and LSCO have essentially the same single electron-like diamond-shaped Fermi surface at p > p* (refs 19,20), Bi2212 has a very different Fermi surface, consisting of two sheets, one of which is also diamond-like at p > 0.22, but the other is much more circular21 (see Supplementary Section 1). Moreover, the structural, magnetic and superconducting properties of Bi2212 are very different to those of Nd-LSCO and LSCO: a stronger 2D character, a larger gap to spin excitations, no spin-density-wave order above p ~ 0.1 and a much higher superconducting Tc.

We measured the resistivity of Bi2212 at p = 0.23 by sup-pressing superconductivity with a magnetic field of up to 58 T. At p = 0.23, the system is just above its pseudogap critical point (p* = 0.22 (ref. 22); see Supplementary Section 2). Our data are shown in Fig. 2. The raw data at H = 55 T reveal a perfectly linear T dependence of ρ(T) down to the lowest accessible temperature (Fig. 1a). Correcting for the magnetoresistance (see Methods and Supplementary Section 3), as was done for LSCO (ref. 8), we find that the T-linear dependence of ρ(T) seen in Bi2212 at H = 0 from T ~ 120 K down to Tc simply continues to low temperature, with the same slope A1 = 0.62 ± 0.06 μ Ω cm K−1 (Fig. 2b). Measured per CuO2 plane, this gives A1

□ ≡ A1/d = 8.0 ± 0.9 Ω K−1, where d is the (aver-age) separation between CuO2 planes. Remarkably, this is the same value, within error bars, as measured in Nd-LSCO at p = 0.24, where A1

□ = 7.4 ± 0.8 Ω K−1 (see Table 1).The observation of T-linear resistivity in those two cuprates

shows that it is robust against changes in the shape, topology and

Universal T-linear resistivity and Planckian dissipation in overdoped cupratesA. Legros1,2, S. Benhabib3, W. Tabis3,4, F. Laliberté" "1, M. Dion1, M. Lizaire1, B. Vignolle3, D. Vignolles" "3, H. Raffy5, Z. Z. Li5, P. Auban-Senzier5, N. Doiron-Leyraud1, P. Fournier1,6, D. Colson2, L. Taillefer" "1,6* and C. Proust" "3,6*

NATURE PHYSICS | VOL 15 | FEBRUARY 2019 | 142–147 | www.nature.com/naturephysics142

Remarkable recent observation of ‘Planckian’ strange metal transportin cuprates, pnictides, magic-angle graphene, and ultracold atoms: theresistivity is associated with a universal scattering time ⇡ ~/(kBT ).

<latexit sha1_base64="96bcgyqY5hBw14jhFAnjyxjijQk=">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</latexit>

REPORT◥

QUANTUM SIMULATION

Bad metallic transport in a cold atomFermi-Hubbard systemPeter T. Brown1, Debayan Mitra1, Elmer Guardado-Sanchez1, Reza Nourafkan2,Alexis Reymbaut2, Charles-David Hébert2, Simon Bergeron2, A.-M. S. Tremblay2,3,Jure Kokalj4,5, David A. Huse1, Peter Schauß1*, Waseem S. Bakr1†

Strong interactions in many-body quantum systems complicate the interpretation ofcharge transport in such materials. To shed light on this problem, we study transport in aclean quantum system: ultracold lithium-6 in a two-dimensional optical lattice, a testingground for strong interaction physics in the Fermi-Hubbard model. We determine thediffusion constant by measuring the relaxation of an imposed density modulation andmodeling its decay hydrodynamically. The diffusion constant is converted to a resistivity byusing the Nernst-Einstein relation. That resistivity exhibits a linear temperaturedependence and shows no evidence of saturation, two characteristic signatures of a badmetal. The techniques we developed in this study may be applied to measurements ofother transport quantities, including the optical conductivity and thermopower.

In conventional materials, charge is carriedby quasiparticles and conductivity is under-stood as a current of these charge carriersdeveloped in response to an external field.For the conductivity to be finite, the charge

carriers must be able to relax their momentumthrough scattering. The Boltzmann kinetic equa-tion in conjunction with Fermi liquid theoryprovides a detailed description of transport inconventionalmaterials, including two trademarksof resistivity. The first is the Fermi liquid predictionthat the temperature (T)–dependent resistivity r(T)should scale like T2 at low temperature (1). Thesecond is that the resistivity should not exceedamaximum value rmax, obtained from the Druderelation assuming the Mott-Ioffe-Regel (MIR)limit, which states that the mean free path of aquasiparticle cannot be less than the lattice spacing(2, 3). This resistivity bound itself is sometimesreferred to as the MIR limit.Strong interactions can, however, lead to a

breakdown of Fermi liquid theory. One signal ofthis breakdown is anomalous scaling of r withtemperature, including the linear scaling ob-served in the strange metal state of the cuprates(4) and other anomalous scalings in d- and f-electron materials (5). Another is the violation ofthe resistivity bound r < rmax, which is observed

in a wide variety of materials (6). Additionally,interactions may lead to a situation where themomentum relaxation rate alone does not de-termine the conductivity, in contrast to thesemiclassical Drude formula, generalizationsof which hold for a large class of systems calledcoherent metals (7). Approaches introduced tounderstand these anomalous behaviors includehidden Fermi liquids (8), marginal Fermi liquids(9), proximity to quantum critical points (10) andassociated holographic approaches (11), andmany numerical studies of model systems,most notably the Hubbard model (12) and thet − J model (13).Disentangling strong interaction physics from

other effects, such as impurities and electron-phonon coupling, is difficult in real materials.Cold atom systems are free of these complica-tions, but transport experiments are challengingbecause of the finite and isolated nature of thesesystems. Most fermionic charge transport ex-periments have focused on studying either massflow through optically structured mesoscopicdevices (14–17) or bulk transport in lattice sys-tems (18–22). In this study, we explored bulktransport in a Fermi-Hubbard system by study-ing charge diffusion, which is a microscopicprocess related to conductivity through theNernst-Einstein equation s = ccD, where D isthe diffusion constant and cc ¼ @n

@m

! "jT is the

compressibility. This requires only the assumptionof linear response and the absence of thermo-electric coupling (23) anddoes not rest on assump-tions concerning quasiparticles.We realized the two-dimensional Fermi-Hubbard

model by using a degenerate spin-balancedmix-ture of two hyperfine ground states of 6Li in anoptical lattice (24). Our lattice beams producea harmonic trapping potential, which leads to avarying atomic density in the trap. To obtain a

system with uniform density, we flatten ourtrapping potential over an elliptical region ofmean diameter 30 sites by using a repulsivepotential created with a spatial light modulator.We superimpose an additional sinusoidal po-tential that varies slowly along one direction ofthe lattice with a controllable wavelength (Fig. 1,A and B). By adiabatically loading the gas intothese potentials, we prepare a Hubbard systemin thermal equilibrium with a small-amplitude(typically 10%) sinusoidal density modulation.The average density in the region with the flat-tened potential is the same with and without thesinusoidal potential. Next, we suddenly turn offthe added sinusoidal potential and observe thedecay of the density pattern versus time (Fig. 1,C and D), always keeping the optical lattice atfixed intensity. We measure the density of asingle spin component, hn↑i, by using techniquesdescribed in (24), giving us access to the totaldensity through hni ¼ h2n↑i.Wework at average total densityhni ¼ 0:82ð2Þ.

This value is close to a conjectured quantumcritical point in the Hubbard model (25). Ourlattice depth is 6.9(2) ER, where ER is the latticerecoil energy and ER/h = 14.66 kHz, leading toa tunneling rate of t/h = 925(10) Hz. Here h isPlanck’s constant. We adjust the scatteringlength, as = 1070(10)ao, by working at a mag-netic bias field of 616.0(2) G, in the vicinity ofthe Feshbach resonance centered near 690 G.These parameters lead to an on-site interaction–to–tunneling ratio U/t = 7.4(8), which is in thestrong-interaction regime and close to the valuethatmaximizes antiferromagnetic correlations athalf-filling (26).We observe the decay of the initial sinusoidal

density pattern over a period of a few tunnelingtimes. The short time scale ensures that theobserved dynamics are not affected by the in-homogeneous density outside of the centralflattened region of the trap. To obtain betterstatistics, we apply the sinusoidal modulationalong one dimension and average along theother direction (Fig. 1, A and C). We fit theaverage modulation profile to a sinusoid, wherethe phase and frequency are fixed by the initialpattern (Fig. 2A). The time dependence of theamplitude of the sinusoid quantifies the decayof the density modulation (Fig. 2B). Our experi-mental technique is analogous to that ofHild et al.(27), who studied the decay of a sinusoidallymodulated spin pattern in a bosonic system.The decay of the sinusoidal density pattern

versus the wavelength l of the modulation be-comes consistent with diffusive transport at longwavelengths. In diffusive transport, the ampli-tude of a density pattern at wave vector k = 2p/lwill decay exponentially with time constant t =1/Dk2, where D is the diffusion constant. We ob-serve exponentially decaying amplitudes withdiffusive scaling for wavelengths longer than15 sites. However, the decay curves are flat atearly times, showing clear deviation from ex-ponential decay. For short wavelengths, we ob-serve deviations from diffusive behavior in theform of underdamped oscillations, which can be

RESEARCH

Brown et al., Science 363, 379–382 (2019) 25 January 2019 1 of 4

1Department of Physics, Princeton University, Princeton, NJ08544, USA. 2Département de Physique, Institut Quantique,and Regroupement Québécois sur les Matériaux de Pointe,Université de Sherbrooke, Sherbrooke, Québec J1K 2R1,Canada. 3Canadian Institute for Advanced Research, Toronto,Ontario M5G 1Z8, Canada. 4Faculty of Civil and GeodeticEngineering, University of Ljubljana, SI-1000 Ljubljana,Slovenia. 5Jožef Stefan Institute, Jamova 39, SI-1000Ljubljana, Slovenia.*Present address: Department of Physics, University of Virginia,Charlottesville, VA 22904, USA.†Corresponding author. Email: wbakr@princeton.edu

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acrossoverto

aconven-

tionalFermiliquid

belowan

emergentlow

-energyscale,

Tcoh ,

atw

hichcoherent

electronicquasiparticles

emerge

asw

ell-defined

excitations.Even

more

strikingexam

plesof

NFL

behaviorare

observedin

thehole-doped

cuprates[6,7]

andcertain

quantumcriticalheavy-ferm

ioncom

pounds[8]w

herethe

incoherentfeaturesappear

tosurvive

down

tothe

lowest

measurable

temperatures,i.e.

Tcoh!

0,when

superconduc-tivity

issuppressedexternally.In

theincoherentregim

e,allofthese

materials

inspite

ofbeingm

icroscopicallydistincthave

aresistivity,

⇢(T

)⇠

T,and

exhibitanum

berof

anomalous

features[8–11]

clearlyindicating

theabsence

ofsharp

elec-tronic

quasiparticles.Instrongly

interactingnon-quasiparticle

systems,ithasbeen

conjectured[12]thattransport“scattering

rates”(�)satisfy

auniversal‘Planckian’bound

�.

O(k

BT/~)

atatem

peratureT

.Itis

howeverw

orthnoting

thatina

NFL

thereisin

generalnocleardefinition

for�;the

scatteringrates

definedthrough

di↵erentmeasurem

ents,suchas

dcand

opti-calconductivity,need

notbeidentical[13].

One

ofthem

ostsurprising

aspectsofincoherenttransportin

thesesystem

s,inspite

ofthesesubtleties,is

that�

extractedfrom

thedc

resis-tivity

(througha

procedurespecified

inR

ef.[14])appears

tosatisfy

auniversalform

�=

Ck

BT/~

with

Ca

numberoforder

1[14,15].

Recent

experiments

[16,17]

havereported

thediscovery

ofacorrelation

driveninsulatoratfractionalfillings

(with

re-spect

toa

fullyfilled

isolatedband)

inm

agic-anglebilayer

graphene(M

AB

LG).

InM

AB

LG,

therelative

rotationbe-

tween

two

sheetsofgraphene

generatesa

moire

pattern(Fig.

1a)w

itha

periodicitythat

ism

uchlarger

thanthe

underly-ing

interatomic

distancesin

graphene.The

theoreticallyesti-

mated

electronicbandw

idth,W

,isstronglyrenorm

alizednear

thesesm

allmagic-angles[18–20];then

thestrength

ofthetyp-

icalCoulom

binteractions,

U,becom

esatleastcom

parableto

(ifnot

greaterthan)

thebandw

idth,U&

W.

Investigatingthe

propertiesof

MA

BLG

asa

functionof

temperature

andcarrier

densityw

ithunprecedented

tunabilityin

acontrolled

setupcan

leadto

newinsights

intothe

natureof

electronictransportin

otherlow-dim

ensionalstronglycorrelated

metal-

licsystem

s.For

ourexperim

ents,w

ehave

fabricatedm

ultiplehigh-

qualityencapsulated

MA

BLG

devices(see

Fig.1a)

usingthe

‘tearand

stack’technique

[21,22].A

sreported

earlier[16,

22],w

eobtain

band-insulatorsw

itha

largegap

nearn⇡±

ns ,w

heren

isthe

carrierdensity

tunedexternally

byapplying

agate

voltageand

ns

correspondsto

fourelectrons

permoire

unitcell.O

nthe

otherhand,correlationdriven

in-sulators

with

much

smaller

gap-scales[16,

23]appear

nearn⇡±

ns /2;

we

denotethese

fillingsas⌫=±

2from

nowon

(thefully

filledband

correspondsto⌫=+

4).D

op-ing

away

fromthese

correlatedinsulators

byan

additionalam

ount,�,with

holes(⌫=±

2��)and

electrons(⌫=±

2+�)

leadsto

superconductivity(SC

)[17,23].

Thesuperconduct-

ingtransition

temperature

(Tc )m

easuredrelative

tothe

Fermi-

energy("

F ),asinferred

fromlow

temperature

quantumoscil-

lationsm

easurements

[17],ishigh,w

iththe

largestvalueof

(Tc /"

F )⇠0.07�

0.08,indicatingstrong

couplingsupercon-

ductivity[17].A

schematic⌫�

Tphase-diagram

forMA

BLG

isshow

nin

Fig.1b.In

thisw

ork,we

investigatethe

transportphenomenology

ofthe

metallic

statesin

MA

BLG

asa

functionof

increasingtem

peratureovera

largerange

ofexternallytuned

fillings.We

haveanalyzed

thetem

peraturedependence

ofthelongitudinal

DC

resistivity,⇢(

T),for

anum

berof

devices(M

A1

-M

A6)

overa

wide

rangeof

fillings,theresults

ofw

hichappear

tobe

qualitativelysim

ilaracross

devices.In

orderto

highlightthe

universalaspectsof

thebehavior,both

within

andacross

di↵erentsamples,w

eshow

thetem

peraturedependenttraces

of⇢(T

)for

arange

ofdi↵erent

⌫in

two

devicesM

A1

andM

A4

inFig.1c

andFig.1d

respectively.Therange

of⌫

chosenforthis

purposeis

shown

asa

color-barinFig.1b;see

caption

arXiv:1901.03710v1 [cond-mat.str-el] 11 Jan 2019

18

Table 1 | Slope of T-linear resistivity and Planckian limit in seven materials.

Material n (1027 m-3)

m*(m0)

A1 / d (! / K)

h / (2e2 TF)(! / K)

⍺

Bi2212 p = 0.23 6.8 8.4 ± 1.6 8.0 ± 0.9 7.4 ± 1.4 1.1 ± 0.3

Bi2201 p ~ 0.4 3.5 7 ± 1.5 8 ± 2 8 ± 2 1.0 ± 0.4

LSCO p = 0.26 7.8 9.8 ± 1.7 8.2 ± 1.0 8.9 ± 1.8 0.9 ± 0.3

Nd-LSCO p = 0.24 7.9 12 ± 4 7.4 ± 0.8 10.6 ± 3.7 0.7 ± 0.4

PCCO x = 0.17 8.8 2.4 ± 0.1 1.7 ± 0.3 2.1 ± 0.1 0.8 ± 0.2

LCCO x = 0.15 9.0 3.0 ± 0.3 3.0 ± 0.45 2.6 ± 0.3 1.2 ± 0.3

TMTSF P = 11 kbar 1.4 1.15 ± 0.2 2.8 ± 0.3 2.8 ± 0.4 1.0 ± 0.3

Table 1 | Slope of T-linear resistivity vs Planckian limit in seven materials.

Comparison of the measured slope of the T-linear resistivity in the T = 0 limit,

A1 , with the value predicted by the Planckian limit (Eq. 1; penultimate column),

for four hole-doped cuprates (Bi2212, Bi2201, LSCO and Nd-LSCO), two

electron-doped cuprates (PCCO and LCCO) and the organic conductor

(TMTSF)2PF6 , as discussed in the text (and Supplementary Information).

The ratio α of the experimental value, A1☐ = A1 / d, over the predicted value,

is given in the last column. Although A1☐ varies by a factor 5, the ratio m* / n

(~1/TF) is seen to vary by the same amount, so that α = 1.0 in all cases,

within error bars.

18

Table 1 | Slope of T-linear resistivity and Planckian limit in seven materials.

Material n (1027 m-3)

m*(m0)

A1 / d (! / K)

h / (2e2 TF)(! / K)

⍺

Bi2212 p = 0.23 6.8 8.4 ± 1.6 8.0 ± 0.9 7.4 ± 1.4 1.1 ± 0.3

Bi2201 p ~ 0.4 3.5 7 ± 1.5 8 ± 2 8 ± 2 1.0 ± 0.4

LSCO p = 0.26 7.8 9.8 ± 1.7 8.2 ± 1.0 8.9 ± 1.8 0.9 ± 0.3

Nd-LSCO p = 0.24 7.9 12 ± 4 7.4 ± 0.8 10.6 ± 3.7 0.7 ± 0.4

PCCO x = 0.17 8.8 2.4 ± 0.1 1.7 ± 0.3 2.1 ± 0.1 0.8 ± 0.2

LCCO x = 0.15 9.0 3.0 ± 0.3 3.0 ± 0.45 2.6 ± 0.3 1.2 ± 0.3

TMTSF P = 11 kbar 1.4 1.15 ± 0.2 2.8 ± 0.3 2.8 ± 0.4 1.0 ± 0.3

Table 1 | Slope of T-linear resistivity vs Planckian limit in seven materials.

Comparison of the measured slope of the T-linear resistivity in the T = 0 limit,

A1 , with the value predicted by the Planckian limit (Eq. 1; penultimate column),

for four hole-doped cuprates (Bi2212, Bi2201, LSCO and Nd-LSCO), two

electron-doped cuprates (PCCO and LCCO) and the organic conductor

(TMTSF)2PF6 , as discussed in the text (and Supplementary Information).

The ratio α of the experimental value, A1☐ = A1 / d, over the predicted value,

is given in the last column. Although A1☐ varies by a factor 5, the ratio m* / n

(~1/TF) is seen to vary by the same amount, so that α = 1.0 in all cases,

within error bars.

A. Legros, S. Benhabib, W. Tabis, F. Laliberté, M. Dion, M. Lizaire, B. Vignolle, D. Vignolles, H. Raffy, Z. Z. Li, P. Auban-Senzier, N. Doiron-Leyraud, P. Fournier, D. Colson, L. Taillefer, and C. Proust, Nature Physics 15, 142 (2019)

1

⌧= ↵

kBT

~<latexit sha1_base64="Q0pGo+lkjPvE7fTn3atRPbcYr7A=">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</latexit>

Horizon radius R =2GM

c2

Objects so dense that light is gravitationally bound to them.

Black Holes

In Einstein’s theory, the region inside the black hole horizon is disconnected from

the rest of the universe.

LIGOSeptember 14, 2015

• The ring-down is predicted by General Relativity to happen in a

time8⇡GM

c3⇠ 8 milliseconds. Curiously this happens to equal

~kBTH

: so the ring down can also be viewed as the approach of a

quantum system to thermal equilibrium at the fastest possible rate.!

Black holes

• Black holes have an entropy anda temperature, TH = ~c3/(8⇡GMkB).

• The entropy is proportional totheir surface area.

• They relax to thermal equilib-rium in a Planckian time⇠ ~/(kBTH).

<latexit sha1_base64="CtCuY4XtgqWdISPYGjZpgV6IJPk=">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</latexit><latexit sha1_base64="CtCuY4XtgqWdISPYGjZpgV6IJPk=">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</latexit><latexit sha1_base64="CtCuY4XtgqWdISPYGjZpgV6IJPk=">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</latexit><latexit sha1_base64="CtCuY4XtgqWdISPYGjZpgV6IJPk=">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</latexit>

Black holes

Holography:Quantum black holes “look like” quantum many-particle systems

without quasiparticle excitations, residing “on” the surface of the black hole

J. Maldacena, IJTP 38, 1113 (1999); S.S. Gubser, I.R. Klebanov, and A.M. Polyakov Phys. Lett. B 428, 105 (1998); E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)

• Black holes have an entropy anda temperature, TH = ~c3/(8⇡GMkB).

• The entropy is proportional totheir surface area.

• They relax to thermal equilib-rium in a Planckian time⇠ ~/(kBTH).

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1. Quantum matter with quasiparticles: random matrix model

2. Quantum matter without quasiparticles: the complex SYK model

3. Fluctuations, and the Schwarzian

4. Models of strange metals

5. Einstein-Maxwell theory of charged black holes in AdS space

1. Quantum matter with quasiparticles: random matrix model

2. Quantum matter without quasiparticles: the complex SYK model

3. Fluctuations, and the Schwarzian

4. Models of strange metals

5. Einstein-Maxwell theory of charged black holes in AdS space

• Note: The electron liquid in one dimension and the fractional

quantum Hall state both have quasiparticles; however, the quasi-

particles do not have the same quantum numbers as an electron.

What are quasiparticles ?

• Quasiparticles are additive excitations:The low-lying excitations of the many-body systemcan be identified as a set {n↵} of quasiparticles withenergy "↵

E =P

↵ n↵"↵ +P

↵,� F↵�n↵n� + . . .

In a lattice system ofN sites, this parameterizes the energyof ⇠ e↵N states in terms of poly(N) numbers.

Ordinary metals and quasiparticles• Quasiparticles eventually collide with each other. Such collisions even-

tually leads to thermal equilibration in a chaotic quantum state, but theequilibration takes a long time. In a Fermi liquid, this time diverges as

⌧eq ⇠ ~E3F

U2(kBT )2, as T ! 0,

where U is the strength of interactions, and EF is the Fermi energy.

• Similarly, a quasiparticle model implies a resistivity

⇢ =m⇤

ne21

⌧⇠ U2T 2 with ⌧ ⇠ ⌧eq

• These times are much longer than the‘Planckian time’ ~/(kBT ), which we willfind in systems without quasiparticle excitations.

⌧ ⇠ ⌧eq � ~kBT

, as T ! 0.<latexit sha1_base64="9/VubhQoZ2AU9raguSa8JnhCjTk=">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</latexit>

Ordinary metals and quasiparticles• Quasiparticles eventually collide with each other. Such collisions even-

tually leads to thermal equilibration in a chaotic quantum state, but theequilibration takes a long time. In a Fermi liquid, this time diverges as

⌧eq ⇠ ~E3F

U2(kBT )2, as T ! 0,

where U is the strength of interactions, and EF is the Fermi energy.

• Similarly, a quasiparticle model implies a resistivity

⇢ =m⇤

ne21

⌧⇠ U2T 2 with ⌧ ⇠ ⌧eq

• These times are much longer than the‘Planckian time’ ~/(kBT ), which we willfind in systems without quasiparticle excitations.

⌧ ⇠ ⌧eq � ~kBT

, as T ! 0.<latexit sha1_base64="9/VubhQoZ2AU9raguSa8JnhCjTk=">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</latexit>

Fermions occupying the eigenstates of a N x N random matrix

tij are independent random variables with tij = 0 and |tij |2 = t2

A simple model of a metal with quasiparticles

H =1

(N)1/2

NX

i,j=1

tijc†i cj � µ

X

i

c†i ci

cicj + cjci = 0 , cic†j + c

†jci = �ij

1

N

X

i

c†i ci = Q

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A simple model of a metal with quasiparticles

⇢(!) =� 1

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Feynman graph expansion in tij.., and graph-by-graph average,yields exact equations in the large N limit:

G(⌧) ⌘ �T⌧

Dci(⌧)c

†i (0)

E

G(i!) =1

i! + µ� ⌃(i!), ⌃(⌧) = t2G(⌧)

G(⌧ = 0�) = Q.

G(!) can be determined by solving a quadratic equation.<latexit sha1_base64="TvyYmNq1c/d62y589QXYkoEDoqw=">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</latexit><latexit sha1_base64="TvyYmNq1c/d62y589QXYkoEDoqw=">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</latexit><latexit sha1_base64="TvyYmNq1c/d62y589QXYkoEDoqw=">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</latexit><latexit sha1_base64="TvyYmNq1c/d62y589QXYkoEDoqw=">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</latexit>

A simple model of a metal with quasiparticles

"↵ levelspacing ⇠ 1/N

⇢(!) =� 1

⇡ ImG(!)<latexit sha1_base64="3hkVLGGiMUm+AUx/4pxVADH9CSY=">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</latexit><latexit sha1_base64="3hkVLGGiMUm+AUx/4pxVADH9CSY=">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</latexit><latexit sha1_base64="3hkVLGGiMUm+AUx/4pxVADH9CSY=">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</latexit><latexit sha1_base64="3hkVLGGiMUm+AUx/4pxVADH9CSY=">AAACUnicdZJNbxMxEIad8FVCgVCOXCxSpCCVyBtlaS9IFT0AtyKRtlIcRV7vbGLVHyt7tm202h/Er+HCgf4TxAknDRIgGMnSq3lnNOPHzkqtAjJ23Wrfun3n7r2t+50H2w8fPe4+2TkJrvISxtJp588yEUArC2NUqOGs9CBMpuE0Oz9a+acX4INy9hMuS5gaMbeqUFJgTM26R9w6ZXOw2NnlfuFon3JnYC5e0jec01e88ELWSVPzUjWUm8xd1R9Mw/fou/6mcHfW7bEB209HrxPKBgljB2kaRTpK03RIkwFbR49s4njW/c5zJysTp0otQpgkrMRpLTwqqaHp8CpAKeS5mMMkSisMhL38QpVhLaf1+t4NfRHNnBbOx2ORrrO/N9fChLA0Waw0Ahfhb2+V/Jc3qbA4mNbKlhWClTeDikpTdHQFkebKg0S9jEJIr+LaVC5EJIURdYcHiO9g57ioOcIVXqo8zqlHyjYR1S8e9P/iZBgpDpKPw97h2w20LfKMPCd9kpB9ckjek2MyJpJ8Jl/IN3Ld+tr60Y6/5Ka03dr0PCV/RHv7JzrOtBs=</latexit>

Let "↵ be the eigenvalues of the matrix tij/pN .

The fermions will occupy the lowest NQ eigen-values, upto the Fermi energy EF . The single-particle density of states is⇢(!) = (1/N)

P↵ �(! � "↵), and ⇢0 ⌘ ⇢(! = 0).

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⇢0<latexit sha1_base64="RADCNYt/31JQIG4ZxLzkbIiT/18=">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</latexit><latexit sha1_base64="RADCNYt/31JQIG4ZxLzkbIiT/18=">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</latexit><latexit sha1_base64="RADCNYt/31JQIG4ZxLzkbIiT/18=">AAACI3icdZDPThsxEMa9lJY0/RfCkYtFqNRDFXkTFcgtggtHKhFAyq5WXu8ksfDaK3sWiFZ5FW4VvAu3qhcOvEhPdUKQoGpHsvTT98147C8tlHTI2H2w8mr19Zu12tv6u/cfPn5qrDdPnCmtgIEwytizlDtQUsMAJSo4KyzwPFVwmp4fzP3TC7BOGn2M0wLinI+1HEnB0UtJoxlpI3UGGuvbkZ2YhG0njRZrs7C702XUw6I87HR730JGw6XSIss6Shq/o8yIMveXCMWdG4aswLjiFqVQMKtHpYOCi3M+hqFHzXNwX7MLWbgFxtXiGzP62ZsZHRnrj0a6UJ8PVzx3bpqnvjPnOHF/e3PxX96wxNFeXEldlAhaPC4alYqiofNMaCYtCFRTD1xY6Z9NxYRbLtAnV48c+Fj1GCdVhHCFlzLze6pOuyf1zIf1lAj9P5x02iFrh987rf7+MrYa2SRb5AsJyS7pk0NyRAZEkCtyTW7IbfAjuAt+Br8eW1eC5cwGeVHBwx8XAqTk</latexit><latexit sha1_base64="RADCNYt/31JQIG4ZxLzkbIiT/18=">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</latexit>

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A simple model of a metal with quasiparticles

The grand potential ⌦(T ) at low T is (from the Sommerfeld expansion)

⌦(T )� E0 = N

✓�⇡2

6⇢0T

2 +O(T 4)

◆+ . . .

where ⇢0 ⌘ ⇢(0) is the single particle density of states at the Fermi level.We can also define the many body density of states, D(E), via

Z = e�⌦(T )/T =

Z 1

�1dED(E)e�E/T

The inversion from ⌦(T ) to D(E) has to performed with care (it need not commutewith the 1/N expansion), and we obtain

D(E) ⇠ exp

⇡

r2N⇢0(E � E0)

3

!, E > E0 ,

1

N⌧ ⇢0(E � E0) ⌧ N

andD(E) = 0 for E < E0. This is related to the asymptotic growth of the partitionsof an integer, p(n) ⇠ exp(⇡

p2n/3). Near the lower bound, there are large sample-

to-sample fluctuations due to variations in the lowest quasiparticle energies.<latexit 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A simple model of a metal with quasiparticles

Quasiparticleexcitations withspacing ⇠ 1/N

There are 2N manybody levels with energy

E =NX

↵=1

n↵"↵,

where n↵ = 0, 1. Shownare all values of E for asingle cluster of size

N = 12. The "↵ have alevel spacing ⇠ 1/N .

Many-bodylevel spacing

⇠ 2�N⇠ Nt<latexit sha1_base64="TLjbKtS021mdKXlJRfXvFeutNo8=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=</latexit><latexit sha1_base64="TLjbKtS021mdKXlJRfXvFeutNo8=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=</latexit><latexit sha1_base64="TLjbKtS021mdKXlJRfXvFeutNo8=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=</latexit><latexit sha1_base64="TLjbKtS021mdKXlJRfXvFeutNo8=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF08SwTwgWcLsZDYZMjO7zvQKIeQnvHhQxKu/482/cfJANLGgoajqprsrSqWw6PtfXm5ldW19I79Z2Nre2d0r7h/UbZIZxmsskYlpRtRyKTSvoUDJm6nhVEWSN6LB9cRvPHJjRaLvcZjyUNGeFrFgFJ3UbFuhyC3BTrHkl/0pyA8JFkkJ5qh2ip/tbsIyxTUySa1tBX6K4YgaFEzycaGdWZ5SNqA93nJUU8VtOJreOyYnTumSODGuNJKp+ntiRJW1QxW5TkWxbxe9ifif18owvgxHQqcZcs1mi+JMEkzI5HnSFYYzlENHKDPC3UpYnxrK0EVUcCEsvbxM6mflwC8Hd+elytU8jjwcwTGcQgAXUIEbqEINGEh4ghd49R68Z+/Ne5+15rz5zCH8gffxDUOnj3I=</latexit>

Fermi liquid state: Two-body interactions lead to a scattering timeof quasiparticle excitations from in (random) single-particle eigen-states which diverges as ⇠ T�2 at the Fermi level.

A simple model of a metal with quasiparticlesNow add weak interactions

H =1

(N)1/2

NX

i,j=1

tijc†i cj � µ

X

i

c†i ci +

1

(2N)3/2

NX

i,j,k,`=1

Uij;k` c†i c

†jckc`

Uij;k` are independent random variables with Uij;k` = 0 and |Uij;k`|2 = U2. We

compute the lifetime of a quasiparticle, ⌧↵, in an exact eigenstate ↵(i) of the

free particle Hamitonian with energy "↵. By Fermi’s Golden rule, for "↵ at the

Fermi energy

1

⌧↵= ⇡U

2⇢30

Zd"�d"�d"�f("�)(1� f("�))(1� f("�))�("↵ + "� � "� � "�)

=⇡3U

2⇢30

4T

2

where ⇢0 is the density of states at the Fermi energy, and f(✏) = 1/(e✏/T

+1) is

the Fermi function.<latexit sha1_base64="P2XD/1G9epCNe0KQg6JmiM9I8aY=">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</latexit>

1. Quantum matter with quasiparticles: random matrix model

2. Quantum matter without quasiparticles: the complex SYK model

3. Fluctuations, and the Schwarzian

4. Models of strange metals

5. Einstein-Maxwell theory of charged black holes in AdS space

A. Kitaev, unpublished; S. Sachdev, PRX 5, 041025 (2015)

S. Sachdev and J. Ye, PRL 70, 3339 (1993)

(See also: the “2-Body Random Ensemble” in nuclear physics; did not obtain the large N limit;T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, and S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981))

U↵�;�� are independent random variables with U↵�;�� = 0 and |U↵�;��|2 = U2

N ! 1 yields critical strange metal.<latexit sha1_base64="yQteaZaLoEYlztplY68/SkIu+Hc=">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 UfkMXlKYvKc7JPX5IAMCPPeeR+8j94nP/M/+1/8r+tU39twHpI/zP/2C0qR6Mw=</latexit>

H =1

(2N)3/2

NX

↵,�,�,�=1

U↵�;�� c†↵c

†�c�c� + ✏

X

↵

c†↵c↵

c↵c� + c�c↵ = 0 , c↵c†� + c

†�c↵ = �↵�

Q =1

N

X

↵

c†↵c↵

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The complex SYK model

U U⌃ =<latexit sha1_base64="pk3ZmbMDJBp861UaFugr9daoJp8=">AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==</latexit><latexit sha1_base64="pk3ZmbMDJBp861UaFugr9daoJp8=">AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==</latexit><latexit sha1_base64="pk3ZmbMDJBp861UaFugr9daoJp8=">AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==</latexit><latexit sha1_base64="pk3ZmbMDJBp861UaFugr9daoJp8=">AAAB7nicbVDLSgMxFL3xWeur6tJNsAiuyswgqAuh6MZlRfuAdiiZNNOGJpkhyQhl6Ee4caGIW7/HnX9j+lho64ELh3Pu5d57olRwYz3vG62srq1vbBa2its7u3v7pYPDhkkyTVmdJiLRrYgYJrhidcutYK1UMyIjwZrR8HbiN5+YNjxRj3aUslCSvuIxp8Q6qdl54H1Jrrulslfxgis/CLAjU2B/kZRhjlq39NXpJTSTTFkqiDFt30ttmBNtORVsXOxkhqWEDkmftR1VRDIT5tNzx/jUKT0cJ9qVsniq/p7IiTRmJCPXKYkdmEVvIv7ntTMbX4Y5V2lmmaKzRXEmsE3w5Hfc45pRK0aOEKq5uxXTAdGEWpdQ0YWw9PIyaQQV36v49+fl6s08jgIcwwmcgQ8XUIU7qEEdKAzhGV7hDaXoBb2jj1nrCprPHMEfoM8fCEKPWw==</latexit>

G<latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit>

G<latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit>

G<latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit><latexit sha1_base64="h54fE11dOI0reWNP9NuJ3oUhZUU=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGQb0FPegxAfOAZAmzk95kzOzsMjMrhCVf4MWDIl79JG/+jZPHQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGt1O/9YRK81g+mHGCfkQHkoecUWOl+l2vWHLLbuXaq1SIJTMQb5mUYIFar/jV7ccsjVAaJqjWHc9NjJ9RZTgTOCl0U40JZSM6wI6lkkao/Wx26IScWaVPwljZkobM1N8TGY20HkeB7YyoGeplbyr+53VSE175GZdJalCy+aIwFcTEZPo16XOFzIixJZQpbm8lbEgVZcZmU7AhrLy8SpqVsueWvfpFqXqziCMPJ3AK5+DBJVThHmrQAAYIz/AKb86j8+K8Ox/z1pyzmDmGP3A+fwC1Gozc</latexit>

The complex SYK model

S. Sachdev and J. Ye, PRL 70, 3339 (1993)

Feynman graph expansion in U↵�;��, and graph-by-graph average, yieldsexact equations in the large N limit:

G(i!) =1

i! � ✏� ⌃(i!), ⌃(⌧) = �U2G2(⌧)G(�⌧)

G(⌧ = 0�) = Q.<latexit sha1_base64="S4bYhQ+/icXvh6gGGPmW8BXeedU=">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</latexit>